natural deduction


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natural deduction

(logic)
A set of rules expressing how valid proofs may be constructed in predicate logic.

In the traditional notation, a horizontal line separates premises (above) from conclusions (below). Vertical ellipsis (dots) stand for a series of applications of the rules. "T" is the constant "true" and "F" is the constant "false" (sometimes written with a LaTeX \perp).

"^" is the AND (conjunction) operator, "v" is the inclusive OR (disjunction) operator and "/" is NOT (negation or complement, normally written with a LaTeX \neg).

P, Q, P1, P2, etc. stand for propositions such as "Socrates was a man". P[x] is a proposition possibly containing instances of the variable x, e.g. "x can fly".

A proof (a sequence of applications of the rules) may be enclosed in a box. A boxed proof produces conclusions that are only valid given the assumptions made inside the box, however, the proof demonstrates certain relationships which are valid outside the box. For example, the box below labelled "Implication introduction" starts by assuming P, which need not be a true proposition so long as it can be used to derive Q.

Truth introduction:

- T

(Truth is free).

Binary AND introduction:

----------- | . | . | | . | . | | Q1 | Q2 | ----------- Q1 ^ Q2

(If we can derive both Q1 and Q2 then Q1^Q2 is true).

N-ary AND introduction:

---------------- | . | .. | . | | . | .. | . | | Q1 | .. | Qn | ---------------- Q1^..^Qi^..^Qn

Other n-ary rules follow the binary versions similarly.

Quantified AND introduction:

--------- | x . | | . | | Q[x] | --------- For all x . Q[x]

(If we can prove Q for arbitrary x then Q is true for all x).

Falsity elimination:

F - Q

(Falsity opens the floodgates).

OR elimination:

P1 v P2 ----------- | P1 | P2 | | . | . | | . | . | | Q | Q | ----------- Q

(Given P1 v P2, if Q follows from both then Q is true).

Exists elimination:

Exists x . P[x] ----------- | x P[x] | | . | | . | | Q | ----------- Q

(If Q follows from P[x] for arbitrary x and such an x exists then Q is true).

OR introduction 1:

P1 ------- P1 v P2

(If P1 is true then P1 OR anything is true).

OR introduction 2:

P2 ------- P1 v P2

(If P2 is true then anything OR P2 is true). Similar symmetries apply to ^ rules.

Exists introduction:

P[a] ------------- Exists x.P[x]

(If P is true for "a" then it is true for all x).

AND elimination 1:

P1 ^ P2 ------- P1

(If P1 and P2 are true then P1 is true).

For all elimination:

For all x . P[x] ---------------- P[a]

(If P is true for all x then it is true for "a").

For all implication introduction:

----------- | x P[x] | | . | | . | | Q[x] | ----------- For all x . P[x] -> Q[x]

(If Q follows from P for arbitrary x then Q follows from P for all x).

Implication introduction:

----- | P | | . | | . | | Q | ----- P -> Q

(If Q follows from P then P implies Q).

NOT introduction:

----- | P | | . | | . | | F | ----- / P

(If falsity follows from P then P is false).

NOT-NOT:

//P --- P

(If it is not the case that P is not true then P is true).

For all implies exists:

P[a] For all x . P[x] -> Q[x] ------------------------------- Q[a]

(If P is true for given "a" and P implies Q for all x then Q is true for a).

Implication elimination, modus ponens:

P P -> Q ---------- Q

(If P and P implies Q then Q).

NOT elimination, contradiction:

P /P ------ F

(If P is true and P is not true then false is true).
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One early attempt to address this problem was undertaken by Andrews, who defined a procedure, later improved by Miller, for converting the logical structures of his proof system to a natural deduction proof style (Miller 1984; Andrews 1981).
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